The presence of different discontinuities requires the use of a monotonic scheme to avoid undershoots and overshoots. It is well known that using \(Q_0\)  finite elements in combination with LLF and explicit time integration that complies with the monotonicity constraints discussed in
[1]}, [2]}, [3]} guarantees the monotonicity of the solution. Hence, a way to obtain monotonic results is
to project the numerical solution onto the \(Q_0\)  subspace for each element in which a suitable jump indicator exceeds a certain threshold.
Similar projections onto low order components of the solution are also used in several monotonization approaches, see e.g. [4]}.
However, since in the proposed scheme only the density is treated in a full explicit fashion, in order to avoid an excessive complication in the
structure of the resulting method we choose to apply this \(Q_0\)  projection strategy only for the density variable, without introducing monotonization for the velocity and the pressure. While we are aware that this is not sufficient to guarantee full monotonicity, the derivation of a fully monotonic IMEX scheme goes beyond the scope of this work and we do not investigate this issue further here.
Therefore, the results in this Section are to be interpreted merely as a first stress test of the proposed scheme at higher Mach number values.
